<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>Ext(Module,Module) -- total Ext module</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Ext^__Z__Z_lp__Coherent__Sheaf_cm__Coherent__Sheaf_rp.html">next</a> | <a href="___Ext.html">previous</a> | <a href="___Ext^__Z__Z_lp__Coherent__Sheaf_cm__Coherent__Sheaf_rp.html">forward</a> | <a href="___Ext.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>Ext(Module,Module) -- total Ext module</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>Ext(M,N)</tt></div>
</dd></dl>
</div>
</li>
<li><span>Scripted functor: <a href="___Ext.html" title="compute an Ext module">Ext</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, <span>an <a href="___Ideal.html">ideal</a></span> or <span>a <a href="___Ring.html">ring</a></span></span></li>
<li><span><tt>N</tt>, <span>a <a href="___Module.html">module</a></span>, <span>an <a href="___Ideal.html">ideal</a></span> or <span>a <a href="___Ring.html">ring</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, the <i>Ext</i> module of <i>M</i> and <i>N</i>, as a multigraded module, with the modules <i>Ext<sup>i</sup>(M,N)</i> for all values of <i>i</i> appearing simultaneously.</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>The modules <tt>M</tt> and <tt>N</tt> should be graded (homogeneous) modules over the same ring.</p>
<p>If <tt>M</tt> or <tt>N</tt> is an ideal or ring, it is regarded as a module in the evident way.</p>
<p>The computation of the total Ext module is possible for modules over the ring <i>R</i> of a complete intersection, according the algorithm of Shamash-Eisenbud-Avramov-Buchweitz. The result is provided as a finitely presented module over a new ring with one additional variable of degree -2,-d for each equation of degree d defining <i>R</i>. The variables in this new ring have degree length 1 more than the degree length of the original ring, i.e., is multigraded, with the degree d part of <i>Ext<sup>n</sup>(M,N)</i> appearing as the degree prepend(-n,d) part of Ext(M,N). We illustrate this in the following example.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y]/(x^3,y^2);</pre>
</td></tr>
<tr><td><pre>i2 : N = cokernel matrix {{x^2, x*y}}
o2 = cokernel | x2 xy |
1
o2 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i3 : H = Ext(N,N);</pre>
</td></tr>
<tr><td><pre>i4 : ring H
o4 = QQ[X , X , x, y]
1 2
o4 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i5 : S = ring H;</pre>
</td></tr>
<tr><td><pre>i6 : H
o6 = cokernel {0, 0} | y2 xy x2 0 0 0 0 0 0 0 0 X_1y X_1x 0 0 0 0 |
{-1, -1} | 0 0 0 y x 0 0 0 0 0 0 0 0 X_1 0 0 0 |
{-1, -1} | 0 0 0 0 0 y x 0 0 0 0 0 0 0 X_1 0 0 |
{-1, -1} | 0 0 0 0 0 0 0 y x 0 0 0 0 0 X_1 0 0 |
{-1, -1} | 0 0 0 0 0 0 0 0 0 y x 0 0 0 0 0 0 |
{-2, -2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y x |
6
o6 : S-module, quotient of S</pre>
</td></tr>
<tr><td><pre>i7 : isHomogeneous H
o7 = true</pre>
</td></tr>
<tr><td><pre>i8 : rank source basis( { -2,-3 }, H)
o8 = 1</pre>
</td></tr>
<tr><td><pre>i9 : rank source basis( { -3 }, Ext^2(N,N) )
o9 = 1</pre>
</td></tr>
<tr><td><pre>i10 : rank source basis( { -4,-5 }, H)
o10 = 4</pre>
</td></tr>
<tr><td><pre>i11 : rank source basis( { -5 }, Ext^4(N,N) )
o11 = 4</pre>
</td></tr>
<tr><td><pre>i12 : hilbertSeries H
-1 -1 2 -1 -2 -2 3 -1 -2 -1 -3 -3 -2 -3 -2 -2 -3 -1
1 + 4T T - 3T - 8T + T T + 2T + 4T T - 4T T - 2T T + 5T + 4T T - 2T T - 2T T
0 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1
o12 = --------------------------------------------------------------------------------------------------------
-2 -2 -2 -3 2
(1 - T T )(1 - T T )(1 - T )
0 1 0 1 1
o12 : Expression of class Divide</pre>
</td></tr>
<tr><td><pre>i13 : hilbertSeries(H,Order=>11)
-1 -1 -2 -3 -2 -2 -3 -4 -4 -6 -3 -3
o13 = 1 + 2T + 4T T + T T + 4T T + 4T T + T T + 2T T +
1 0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-4 -5 -5 -7 -6 -9 -4 -4 -5 -6 -6 -8 -7 -10
4T T + 4T T + T T + 2T T + 2T T + 4T T + 4T T +
0 1 0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-8 -12 -5 -5 -6 -7 -7 -9 -8 -11 -9 -13 -10 -15
T T + 2T T + 2T T + 2T T + 4T T + 4T T + T T
0 1 0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-6 -6 -7 -8 -8 -10 -9 -12 -10 -14 -11 -16
+ 2T T + 2T T + 2T T + 2T T + 4T T + 4T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-12 -18 -7 -7 -8 -9 -9 -11 -10 -13 -11 -15
T T + 2T T + 2T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-12 -17 -13 -19 -14 -21 -8 -8 -9 -10 -10 -12
4T T + 4T T + T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-11 -14 -12 -16 -13 -18 -14 -20 -15 -22 -16 -24
2T T + 2T T + 2T T + 4T T + 4T T + T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-9 -9 -10 -11 -11 -13 -12 -15 -13 -17 -14 -19
2T T + 2T T + 2T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-15 -21 -16 -23 -17 -25 -18 -27 -10 -10 -11 -12
2T T + 4T T + 4T T + T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-12 -14 -13 -16 -14 -18 -15 -20 -16 -22 -17 -24
2T T + 2T T + 2T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-18 -26 -19 -28 -20 -30
4T T + 4T T + T T
0 1 0 1 0 1
o13 : ZZ[T , T ]
0 1</pre>
</td></tr>
</table>
<p>The result of the computation is cached for future reference.</p>
</div>
</div>
</div>
</body>
</html>
